Match each integrand in List I with its integral in List II (a > 0; add C to each).

Match each integrand in List I with its…

Question 50

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Match each integrand in List I with its integral in List II ( $a > 0$ ; add $C$ to each).

List I	List II
A. $\frac{1}{x^2 + a^2}$	I. $\sin^{-1}\frac{x}{a}$
B. $\frac{1}{x^2 - a^2}$	II. $\frac{1}{a}\tan^{-1}\frac{x}{a}$
C. $\frac{1}{\sqrt{a^2 - x^2}}$	III. $\log\left\|x + \sqrt{x^2 + a^2}\right\|$
D. $\frac{1}{\sqrt{x^2 + a^2}}$	IV. $\frac{1}{2a}\log\left\|\frac{x - a}{x + a}\right\|$

A-IV, B-II, C-I, D-IIICorrect answerYour answerA-II, B-IV, C-III, D-ICorrect answerYour answerA-IV, B-II, C-III, D-ICorrect answerYour answerA-II, B-IV, C-I, D-IIICorrect answerYour answer

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CorrectNot quiteAnswer D

Correct answer: D

Each can be checked by differentiating: for example $\frac{d}{dx}\sin^{-1}\frac{x}{a} = \frac{1}{\sqrt{a^2 - x^2}}$ .
A plus sign without a root gives $\tan^{-1}$ ; a minus sign gives a logarithm of a ratio; the roots give $\sin^{-1}$ and the logarithm with $x + \sqrt{\;}$ .

Concept: The table of particular integrals
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsIntegrals of particular functions

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Ten questions from Indefinite Integrals, at your own pace.

Practise this chapter →

Ten questions from Indefinite Integrals, at your own pace.

Question 50

Explanation